I am reading the Landau & Lifshitz on mechanics to understand how we find the free particle Lagrangian, and there are some things that I don't understand.
First, he defines an inertial frame as the following. He says that an inertial frame is a frame in which space is homegenous and isotropic, and time is homogenous.
What does that exactly mean?
I will only take the example of homogenous space for the next part of my question.
I have read on various topic on this website that homogenity of space means if I change $q \rightarrow q+q_0$ in the Lagrangian, then it will be unchanged.
Im ok with this definition but at this moment the book didn't talk about this at all.
So I guess that they want to say that the equation of motion will be unchanged if the position of the particle is changed from $q$ to $q+a$ in the frame (I talk about the differential equation of motion, and not the solution $q(t)$ that will change because of different initial condition).
First question : Am I right with this definition?
Ok, then imagine that I have a particle at rest (so, it doesn't feel any force). I take an accelerated frame in the $x$ direction at constant acceleration.
The equation of motion in this frame of the particle will be : $\ddot{x}=a$
If I use the definition of an inertial frame given by Landau, I would find that this frame is an inertial frame : if I change $x \rightarrow x+x_0$, the equation of motion will be unchanged. And I can do the same for time : $ t \rightarrow t+t_0$, and for isotropy : I have the homogeneity everywhere so it is necesseraly isotropic.
Where is my mistake? Because of course an accelerated frame is not inertial. What did I misunderstood in Landau's Book?
[edit]: I just read from the first answer from here Mechanics Landau Galilean Principle what is the definition of homogenous space, time and isotropic space.
But even with this definition I don't see why an accelerated frame will not be inertial according to Landau definition of an inertial frame.
